The invariance of the Artin root number under induction can be proved without special effort. In fact if one develops the properties of Artin Lfunctions in the usual way, then the inductivity of the root number is obvious. In barest outline the argument is as follows. First one proves the inductivity of Artin L-functions themselves, and then one combines Brauer’s induction theorem with the anal...

In this note we confirm two conjectural supercongruences of double sums binomial coefficients due to El Bachraoui.

Let G be a connected, simply-connected, simple algebraic group over C. We fix a Borel subgroup B of G and a maximal torus T ⊂ B. We denote their Lie algebras by g, b, h respectively. Let P+ ⊂ h be the set of dominant integral weights. For any λ ∈ P+, let V (λ) be the finite dimensional irreducible g-module with highest weight λ. We fix a positive integer l and let Rl(g) be the free Z-module wit...

A theorem of Kazhdan and Lusztig establishes an equivalence between the category $G(\mathcal{O})$-integrable representations Kac--Moody algebra $\widehat{\mathfrak{g}}\_{-\kappa}$ at a negative level $-\kappa$ $\operatorname{Rep}\_q(G)$ (algebraic) “big” (a.k.a. Lusztig's) quantum group. In this paper we propose conjecture that describes Iwahori-integrable Kac–Moody modules. The corresponding o...

The optimal bidding strategy for a generation company is a complex decision making problem involving numerous uncertainties. Generally, all market participants attempt to refine their strategies to earn greater profit. Success in the market requires not only successful forecasting of demand and other market conditions but also anticipating rival behaviours. The main objective of this paper is t...